Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-3x-6y &= -9 \\ -4x+4y &= 1\end{align*}$
Explanation: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $2$ and the bottom equation by $3$ $\begin{align*}-6x-12y &= -18\\ -12x+12y &= 3\end{align*}$ Add the top and bottom equations. $-18x = -15$ Divide both sides by $-18$ and reduce as necessary. $x = \dfrac{5}{6}$ Substitute $\dfrac{5}{6}$ for $x$ in the top equation. $-3( \dfrac{5}{6})-6y = -9$ $-\dfrac{5}{2}-6y = -9$ $-6y = -\dfrac{13}{2}$ $y = \dfrac{13}{12}$ The solution is $\enspace x = \dfrac{5}{6}, \enspace y = \dfrac{13}{12}$.